Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN

Gauge Equivariant Convolutional Networks and the Icosahedral CNN

The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.Comment: Proceedings of the International Conference on Machine Learning (ICML), 201

Video-based, student tutor- versus faculty staff-led ultrasound course for medical students - a prospective randomized study

Background Ultrasound education is propagated already during medical school due to its diagnostic importance. Courses are usually supervised by experienced faculty staff (FS) with patient bedside examinations or students among each other but often overbooked due to limited FS availability. To overcome this barrier, use of teaching videos may be advantageous. Likewise, peer teaching concepts solely with trained student tutors have shown to be feasible and effective. The aim was to evaluate 1) objective learning outcomes of a combined video-based, student-tutor (ViST) as compared to a FS-led course without media support, 2) acceptance and subjective learning success of the videos. Methods Two ultrasound teaching videos for basic and advanced abdominal ultrasound (AU) and transthoracic echocardiography (TTE) were produced and six students trained as tutors. Fourth-year medical students (N = 96) were randomized to either the ViST- or FS course (6 students per tutor). Learning objectives were defined equally for both courses. Acquired practical basic and advanced ultrasound skills were tested in an objective structured clinical examination (OSCE) using modified validated scoring sheets with a maximum total score of 40 points. Acceptance and subjective learning success of both videos were evaluated by questionnaires based on Kirkpatrick's evaluation model with scale-rated closed and open questions. Results 79 of 96 medical students completed the OSCE and 77 could be finally analyzed. There was no significant difference in the mean total point score of 31.3 in the ViST (N = 42) and 32.7 in the FS course (N = 35, P = 0.31) or in any of the examined basic or advanced ultrasound skill subtasks. Of the 42 ViST participants, 29 completed the AU and 27 the TTE video questionnaire. Acceptance and subjective learning success of both videos was rated positively in 14-52% and 48-88% of the rated responses to each category, respectively. Attendance of either the student or faculty tutor was deemed necessary in addition to the videos. Conclusions A ViST versus FS teaching concept was able to effectively teach undergraduate students in AU and TTE, albeit acceptance of the teaching videos alone was limited. However, the ViST concept has the potential to increase course availability and FS resource allocation

Gamma-Ray Emission from Two Blazars Behind the Galactic Plane: B2013+370 & B2023+336

B2013+370 and B2023+336 are two blazars at low-galactic latitude that were previously proposed to be the counterparts for the EGRET unidentified sources, 3EG J2016+3657 and 3EG J2027+3429. Gamma-ray emission associated with the EGRET sources has been detected by the Fermi Gamma-ray Space Telescope, and the two sources, 1FGL J2015.7+3708 and 1FGL J2027.6+3335, have been classified as unidentified in the 1-year catalog. This analysis of the Fermi-LAT data collected during 31 months reveals that the 1FGL sources are spatially compatible with the blazars, and are significantly variable, supporting the hypothesis of extragalactic origin for the gamma-ray emission. The gamma-ray light curves are compared with 15 GHz radio light curves from the 40-m telescope at the Owens Valley Radio Observatory (OVRO). Simultaneous variability is seen in both bands for the two blazar candidates. The study is completed with the X-ray analysis of 1FGL J2015.7+3708 using Swift observations that were triggered in August 2010 by a Fermi-detected flare. The resulting spectral energy distribution shows a two-component structure typical of blazars. We also identify a second source in the field of view of 1FGL J2027.6+3335 with similar characteristics to the known LAT pulsars. This study gives solid evidence favoring blazar counterparts for these two unidentified EGRET and Fermi sources, supporting the hypothesis that a number of unidentified gamma-ray sources at low galactic latitudes are indeed of extragalactic origin.Comment: 10 pages, 7 figures, 6 tables, accepted for publication in The Astrophysical Journa

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